{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# p16: Poisson equation on [-1,1]x[-1,1] with u=0 on boundary"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We solve the following Poisson problem\n",
    "$$\n",
    "u_{xx} + u_{yy} = 10\\sin(8x(y-1)), \\qquad -1 < x,y < 1, \\qquad u=0 \\quad \\mbox{on boundary}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format='svg'\n",
    "from chebPy import cheb\n",
    "from numpy import meshgrid,sin,dot,eye,kron,zeros,reshape,linspace\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "from matplotlib.pyplot import figure,subplot,plot,title,axis,xlabel,ylabel,spy\n",
    "from matplotlib import cm\n",
    "from scipy.linalg import solve\n",
    "from scipy.interpolate import interp2d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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     },
     "metadata": {},
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   ],
   "source": [
    "N = 24; D,x = cheb(N); y = x;\n",
    "xx,yy = meshgrid(x[1:N],y[1:N])\n",
    "xx = reshape(xx,(N-1)**2)\n",
    "yy = reshape(yy,(N-1)**2)\n",
    "f = 10*sin(8*xx*(yy-1))\n",
    "D2 = dot(D,D); D2 = D2[1:N,1:N]; I = eye(N-1)\n",
    "L = kron(I,D2) + kron(D2,I)\n",
    "# Plot sparsity pattern\n",
    "figure(figsize=(8,8)), spy(L)\n",
    "# Solve Lu=f\n",
    "u = solve(L,f)\n",
    "# Convert 1-d vectors to 2-d\n",
    "uu = zeros((N+1,N+1)); uu[1:N,1:N] = reshape(u,(N-1,N-1))\n",
    "[xx,yy] = meshgrid(x,y)\n",
    "value = uu[N/4,N/4]\n",
    "\n",
    "# Interpolate to finer mesh just for visualization\n",
    "f = interp2d(x,y,uu,kind='cubic')\n",
    "xxx = linspace(-1.0,1.0,50)\n",
    "uuu = f(xxx,xxx)\n",
    "fig = figure(figsize=(8,8))\n",
    "ax = fig.add_subplot(111, projection='3d')\n",
    "X,Y = meshgrid(xxx,xxx)\n",
    "ax.plot_surface(X,Y,uuu,rstride=1,cstride=1,cmap=cm.jet)\n",
    "title(\"$u(2^{-1/2},2^{-1/2})$=\"+str(value))\n",
    "xlabel(\"x\"); ylabel(\"y\");"
   ]
  }
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